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Theorem dvlog2lem 20015
Description: Lemma for dvlog2 20016. (Contributed by Mario Carneiro, 1-Mar-2015.)
Hypothesis
Ref Expression
dvlog2.s  |-  S  =  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )
Assertion
Ref Expression
dvlog2lem  |-  S  C_  ( CC  \  (  -oo (,] 0 ) )

Proof of Theorem dvlog2lem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dvlog2.s . . . . 5  |-  S  =  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )
2 cnxmet 18298 . . . . . 6  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
3 ax-1cn 8811 . . . . . 6  |-  1  e.  CC
4 1re 8853 . . . . . . 7  |-  1  e.  RR
5 rexr 8893 . . . . . . 7  |-  ( 1  e.  RR  ->  1  e.  RR* )
64, 5ax-mp 8 . . . . . 6  |-  1  e.  RR*
7 blssm 17984 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  1  e.  CC  /\  1  e.  RR* )  ->  (
1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC )
82, 3, 6, 7mp3an 1277 . . . . 5  |-  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC
91, 8eqsstri 3221 . . . 4  |-  S  C_  CC
109sseli 3189 . . 3  |-  ( x  e.  S  ->  x  e.  CC )
113subid1i 9134 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
12 mnfxr 10472 . . . . . . . . . . . 12  |-  -oo  e.  RR*
13 0re 8854 . . . . . . . . . . . 12  |-  0  e.  RR
14 iocssre 10745 . . . . . . . . . . . 12  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  (  -oo (,] 0 )  C_  RR )
1512, 13, 14mp2an 653 . . . . . . . . . . 11  |-  (  -oo (,] 0 )  C_  RR
1615sseli 3189 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  e.  RR )
1713a1i 10 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  0  e.  RR )
184a1i 10 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  RR )
19 elioc2 10729 . . . . . . . . . . . 12  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  (
x  e.  (  -oo (,] 0 )  <->  ( x  e.  RR  /\  -oo  <  x  /\  x  <_  0
) ) )
2012, 13, 19mp2an 653 . . . . . . . . . . 11  |-  ( x  e.  (  -oo (,] 0 )  <->  ( x  e.  RR  /\  -oo  <  x  /\  x  <_  0
) )
2120simp3bi 972 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  <_  0 )
2216, 17, 18, 21lesub2dd 9405 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1  -  0 )  <_  ( 1  -  x ) )
2311, 22syl5eqbrr 4073 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  1  <_  ( 1  -  x
) )
24 ax-resscn 8810 . . . . . . . . . . . 12  |-  RR  C_  CC
2515, 24sstri 3201 . . . . . . . . . . 11  |-  (  -oo (,] 0 )  C_  CC
2625sseli 3189 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  e.  CC )
27 eqid 2296 . . . . . . . . . . 11  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2827cnmetdval 18296 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1 ( abs 
o.  -  ) x
)  =  ( abs `  ( 1  -  x
) ) )
293, 26, 28sylancr 644 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( abs `  (
1  -  x ) ) )
30 0le1 9313 . . . . . . . . . . . 12  |-  0  <_  1
3130a1i 10 . . . . . . . . . . 11  |-  ( x  e.  (  -oo (,] 0 )  ->  0  <_  1 )
3216, 17, 18, 21, 31letrd 8989 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  <_  1 )
3316, 18, 32abssubge0d 11930 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs `  ( 1  -  x ) )  =  ( 1  -  x
) )
3429, 33eqtrd 2328 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( 1  -  x ) )
3523, 34breqtrrd 4065 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  1  <_  ( 1 ( abs 
o.  -  ) x
) )
36 cnmet 18297 . . . . . . . . . 10  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
3736a1i 10 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs  o.  -  )  e.  ( Met `  CC ) )
383a1i 10 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  CC )
39 metcl 17913 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  1  e.  CC  /\  x  e.  CC )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
4037, 38, 26, 39syl3anc 1182 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
41 lenlt 8917 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  ( 1 ( abs 
o.  -  ) x
)  e.  RR )  ->  ( 1  <_ 
( 1 ( abs 
o.  -  ) x
)  <->  -.  ( 1 ( abs  o.  -  ) x )  <  1 ) )
424, 40, 41sylancr 644 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1  <_  ( 1 ( abs  o.  -  ) x )  <->  -.  (
1 ( abs  o.  -  ) x )  <  1 ) )
4335, 42mpbid 201 . . . . . 6  |-  ( x  e.  (  -oo (,] 0 )  ->  -.  ( 1 ( abs 
o.  -  ) x
)  <  1 )
442a1i 10 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs  o.  -  )  e.  ( * Met `  CC ) )
456a1i 10 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  RR* )
46 elbl2 17966 . . . . . . 7  |-  ( ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  1  e.  RR* )  /\  ( 1  e.  CC  /\  x  e.  CC ) )  -> 
( x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4744, 45, 38, 26, 46syl22anc 1183 . . . . . 6  |-  ( x  e.  (  -oo (,] 0 )  ->  (
x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4843, 47mtbird 292 . . . . 5  |-  ( x  e.  (  -oo (,] 0 )  ->  -.  x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 ) )
4948con2i 112 . . . 4  |-  ( x  e.  ( 1 (
ball `  ( abs  o. 
-  ) ) 1 )  ->  -.  x  e.  (  -oo (,] 0
) )
5049, 1eleq2s 2388 . . 3  |-  ( x  e.  S  ->  -.  x  e.  (  -oo (,] 0 ) )
51 eldif 3175 . . 3  |-  ( x  e.  ( CC  \ 
(  -oo (,] 0 ) )  <->  ( x  e.  CC  /\  -.  x  e.  (  -oo (,] 0
) ) )
5210, 50, 51sylanbrc 645 . 2  |-  ( x  e.  S  ->  x  e.  ( CC  \  (  -oo (,] 0 ) ) )
5352ssriv 3197 1  |-  S  C_  ( CC  \  (  -oo (,] 0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696    \ cdif 3162    C_ wss 3165   class class class wbr 4039    o. ccom 4709   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   (,]cioc 10673   abscabs 11735   * Metcxmt 16385   Metcme 16386   ballcbl 16387
This theorem is referenced by:  dvlog2  20016  logtayl  20023  logtayl2  20025  efrlim  20280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-xadd 10469  df-ioc 10677  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-xmet 16389  df-met 16390  df-bl 16391
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